String topology of classifying spaces

Abstract

Let $G$ be a finite group or a compact connected Lie group and let $BG$ be its classifying space. Let $\mathcal{L}BG:=map(S^1,BG)$ be the free loop space of $BG$ i.e. the space of continuous maps from the circle $S^1$ to $BG$. The purpose of this paper is to study the singular homology $H_{\ast }(\mathcal{L}BG)$ of this loop space. We prove that when taken with coefficients in a field the homology of $\mathcal{L}BG$ is a homological conformal field theory. As a byproduct of our main theorem, we get a Batalin-Vilkovisky algebra structure on the cohomology $H^{\ast }(\mathcal{L}BG)$. We also prove an algebraic version of this result by showing that the Hochschild cohomology $H{H}^{\ast }(S_{\ast }(G),S_{\ast }(G))$ of the singular chains of $G$ is a Batalin-Vilkovisky algebra.

53 pages. Section 3 on Props and fields theories rewritten. Section 4 expanded in new sections 4, 5, 6 and 7, to fix orientation problems, finite groups case detailed in section 7. Appendix on signs added. The rest of the sections almost unchanged. Some slight improvements on some results. For example, the BV-algebra is valid over any principal ideal domain

References in corpus (2)

• Higher string topology operations
• Strings and the stable cohomology of mapping class groups

Cited by in corpus (1)

• String topology on Gorenstein spaces